Ant-inspired density estimation via random walks
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Musco, Cameron et al. “Ant-Inspired Density Estimation via Random
Walks.” Proceedings of the National Academy of Sciences 114,
40 (September 2017): 10534–10541 © 2017 National Academy of
Sciences
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http://dx.doi.org/10.1073/PNAS.1706439114
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National Academy of Sciences (U.S.)
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Ant-inspired density estimation via random walks
Cameron Muscoa,1, Hsin-Hao Sua,1, and Nancy A. Lyncha,1,2aComputer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139
This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2016.
Contributed by Nancy A. Lynch, August 14, 2017 (sent for review April 18, 2017; reviewed by Yehuda Afek, Ziv Bar-Joseph, and Amos Korman)
Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical per-spective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understand-ing complex dependencies in the collision probabilities of multi-ple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density esti-mation for robot swarms, and random walk-based sampling for sensor networks.
population density estimation | random walk sampling | network exploration | ant colony algorithms | biological distributed algorithms
T
he ability to sense local population density is an important tool used by many ant species. When a colony of Temnotho-rax ants must relocate to a new nest, scouts search for potential nest sites, assess their quality, and recruit other scouts to high-quality locations. A high enough density of scouts at a poten-tial new nest (a quorum threshold) triggers those ants to decide on the site and transport the rest of the colony there (2). When neighboring colonies of Azteca ants compete for territory, a high relative density of a colony’s ants in a contested area will cause those ants to attack enemies in the area, while a low relative den-sity will cause the colony to retreat (3). Varying densities of har-vester ants successfully performing certain tasks such as foraging or brood care can trigger other ants to switch tasks, maintaining proper worker allocation in the colony (4, 5).It has been shown that ants estimate density in a distributed manner, by measuring encounter rates (2, 6). As ants randomly walk around an area, if they bump into a larger number of other ants, this indicates a higher population density. By track-ing encounters with specific types of ants, for example, successful foragers or enemies, ants can estimate more specific densities. This strategy allows each ant to obtain an accurate density esti-mate and requires very little communication: Ants must simply detect when they collide and do not need to perform any higher-level data aggregation.
Density Estimation on a Grid
We study distributed density estimation from a theoretical per-spective. We model a colony of ants as a set of anonymous agents randomly placed on a 2D grid. Computation proceeds in rounds, with each agent stepping in a random direction in each round.
A collision occurs when two agents reach the same position in the same round, and encounter rate is measured as the number of collisions an agent is involved in during a sequence of rounds divided by the number of rounds. Aside from collision detection, the agents have no other means of communication.
The intuition that encounter rate tracks density is clear. It is easy to show that, for a set of randomly walking agents, the expected encounter rate measured by each agent is exactly the density d —the number of agents divided by the grid size (see Lemma 2). However, it is unclear if encounter rate actually gives a good density estimate, that is, if the estimate is close to its expectation with high probability.
Consider agents positioned not on the grid but on a complete graph. In each round, each agent steps to a uniformly random position, and, in expectation, the number of other agents it col-lides with in this step is d . Since each agent chooses its new loca-tion uniformly at random in each step, collisions are essentially independent between rounds. The agents are effectively taking independent Bernoulli samples with success probability d , and, by a standard Chernoff bound, within O log(1/δ)/d 2 rounds, each obtains a (1 ± ) multiplicative approximation to d with probability 1 − δ.
On the grid graph, the picture is significantly more complex. If two agents are initially located near each other, they are more likely to collide via random walking. After a first collision, due to their proximity, they are likely to collide repeatedly in future rounds. Since the agents are anonymous, they cannot recognize repeat collisions, and, even if they could, it is unclear that it would help. On average, compared with the complete graph,
Significance
Highly complex distributed algorithms are ubiquitous in nature: from the behavior of social insect colonies and bird flocks, to cellular differentiation in embryonic development, to neural information processing. In our research, we study biological computation theoretically, combining a scientific perspective, which seeks to better understand the systems being studied, with an engineering perspective, which takes inspiration from these systems to improve algorithm design. In this work, we focus on the problem of population density estimation in ant colonies, demonstrating that extremely sim-ple algorithms, similar to those used by ants, solve the prob-lem with strong theoretical guarantees and have a number of interesting computational applications.
An extended abstract of this work has been previously published (1).
Author contributions: C.M., H.-H.S., and N.A.L. designed research; C.M., H.-H.S., and N.A.L. performed research; C.M. and H.-H.S. contributed new reagents/analytic tools; and C.M., H.-H.S., and N.A.L. wrote the paper.
Reviewers: Y.A., Tel-Aviv University; Z.B.-J., Carnegie Mellon University; and A.K., French National Center for Scientific Research.
The authors declare no conflict of interest. See QnAs on page 10512.
1C.M., H.S., and N.L. contributed equally to this work.
2To whom correspondence should be addressed. Email: lynch@csail.mit.edu.
This article contains supporting information online atwww.pnas.org/lookup/suppl/doi:10.
INAUGURAL ARTICLE COMPUTER SCIENCES SYSTEMS BIOLOGY
agents collide with fewer individuals and collide multiple times with those individuals that they do encounter, making encounter rates a less reliable estimate of population density.
Mathematically speaking, on a graph with a fast mixing time (7), like the complete graph, each agent’s location is only weakly correlated with its previous locations. This ensures that collisions are also weakly correlated between rounds, and encounter rate serves as a very accurate estimate of density. The grid graph, on the other hand, is slow mixing: Agent positions and hence col-lisions are highly correlated between rounds, lowering the accu-racy of encounter-rate-based estimation.
Results
Surprisingly, despite the high correlation between collisions, we show that encounter rate-based density estimation on the grid is nearly as accurate as on the complete graph. After just O log(1/δ) log log(1/δ) log(1/d )/d 2
rounds, each agent’s encounter rate is a (1 ± ) approximation to d with probability 1 − δ(Theorem 1). This matches performance on the complete graph up to a log log(1/δ) log(1/d ) factor.
Technically, to bound accuracy on the grid, we obtain moment bounds on the number of times that two randomly walking agents collide over a set of rounds (Lemma 5). These bounds also apply to the number of equalizations (returns to origin) of a single walk. While expected random walk hitting times, return times, and collision rates are well studied for many graphs, including grid graphs (7–9), higher moment bounds and high probability results are much less common.
Our moment bounds show that, while the grid graph is slow mixing, it has strong local mixing. That is, random walks tend to spread quickly over a local area and not repeatedly cover the same nodes, making random walk-based density estimation accurate. Significant work has focused on showing that random walk sampling is nearly as good as independent sampling for fast-mixing expander graphs (10, 11). We extend this type of analysis to slowly mixing graphs, showing that strong local mixing is suffi-cient in many applications.
The key to the local mixing property of the grid is an upper bound on the probability that two random walks starting from the same position recollide (or that a single random walk equal-izes) after a certain number of steps (Lemma 3). We show that recollision probability bounds imply collision moment bounds on general graphs, and apply this technique to extend our results to d-dimensional grids, regular expanders, and hypercubes. We dis-cuss applications of our bounds to the task of estimating the size of a social network using random walks (12), obtaining improve-ments over prior work for networks with relatively slow global mixing times but strong local mixing. We also discuss connections to density estimation by robot swarms and random walk-based sensor network sampling (13, 14).
Theoretical Model for Density Estimation
We consider a set of agents populating a 2D torus with A nodes (dimensions√A ×√A). At each time step, each agent has an associated ordered pair position, which gives its coordinates on the torus. We assume that A is large—larger than the area agents traverse over the runtimes of our algorithms. We believe the torus model successfully captures the dynamics of density estima-tion on a surface, while avoiding complicating factors of bound-ary behavior on a finite grid.
Initially, each agent is placed independently at a uniform ran-dom node in the torus. Computation proceeds in discrete, syn-chronous rounds. Each agent updates its position with a step cho-sen uniformly at random from {(0, 1), (0, −1), (1, 0), (−1, 0)} in each round. Of course, in reality, ants do not move via pure ran-dom walk; observed encounter rates seem to actually be lower than predicted by a pure random walk model (6, 15). How-ever, we feel that our model sufficiently captures the highly
ran-dom movement of ants while remaining tractable to analysis and applicable to ant-inspired random walk-based algorithms. Extending our work to more realistic models of ant movement would be an interesting next direction.
Aside from the ability to move in each round, agents can sense the number of agents other than themselves at their posi-tion at the end of each round, formally through the funcposi-tion count (position). We say that two agents collide in round r if they have the same position at the end of the round. Outside of col-lision counting, agents have no means of communication. They are anonymous (cannot uniquely identify each other) and exe-cute identical density estimation routines. A basic illustration of our model is depicted in Fig. 1.
The Density Estimation Problem
Let (n + 1) be the number of agents, and define population den-sity as d def= n/A. Each agent’s goal is to estimate d to (1 ± ) accuracy with probability at least 1 − δ for , δ ∈ (0, 1), that is, to return an estimate ˜d with Ph ˜d ∈ [(1 − )d , (1 + )d ]i≥ 1 − δ. As a technicality, with n + 1 agents, we define d = n/A instead of d = (n + 1)/A, for convenience of calculation. In the natural case, when n is large, the distinction is unimportant.
Local vs. Global Density. The problem described above requires estimating the global population density. We assume that agents are initially distributed uniformly at random on the torus, which is critical for fast global density estimation: When agents are uni-formly distributed, the local density in a small radius around their starting position reflects the global density with good probabil-ity. Of course, in nature, ants are not typically uniformly dis-tributed in the nest or surrounding areas. Additionally, they are often interested in estimating local population densities, e.g., in a new nest site when house-hunting (2) or around a nest entrance
Fig. 1. A basic illustration of our computational model. Each agent (ant) may move to a random adjacent position on the 2D torus in each round (illustrated by the red arrows). A collision occurs when two or more agents are located at the same position. The agents detect collisions through the
count(position) function, which returns the number of other agents at their
current position. In this illustration, position is given as the (x, y) position, with the bottom left corner corresponding to (1, 1). However, the precise convention used is unimportant.
when estimating the number of successful foragers for task allocation (4).
We view our work as a first step toward a theoretical under-standing of density estimation, and we focus on the global density for simplicity. Removing our assumption of uniformly distributed agents and understanding local density estimation are important directions for future work.
Random Walk-Based Density Estimation on the 2D Torus As discussed, the challenge in analyzing random walk-based den-sity estimation on the torus arises from correlations between col-lisions of nearby agents. If we do not restrict agents to random walking, and instead allow each agent to take an arbitrary step in each round, they can avoid collision correlations by splitting into “stationary” and “mobile” groups and counting collisions only between members of different groups. This allows them to essentially simulate independent sampling of grid locations to estimate density. This method is simple to analyze (SI Appendix,
section S1), but it is not “natural” in a biological sense or useful for the applications we present. Further, independent sampling is unnecessary! Algorithm 1 describes a simple random walk-based approach that gives a nearly matching bound.
Our main theoretical result follows; its proof appears at the end of this section, after a number of preliminary lemmas. Throughout our analysis, we take the viewpoint of a single agent executing Algorithm 1.
Theorem 1 (Random Walk Sampling Accuracy Bound).After run-ning for t rounds, assuming t ≤ A, an agent executing Algorithm 1 returns ˜d such that, for any δ > 0, with a probability of ≥1 − δ,
˜
d ∈ [(1 − )d , (1 + )d ] for = Θplog(1/δ) log(2t)/td.In other words, for any , δ ∈ (0, 1) if t = Θ log(1/δ) log log(1/δ) log(1/d )/d 2, ˜dis a (1 ± ) multiplicative estimate of d with a probability of ≥1 − δ.
Theorem 1 focuses on the density estimate of a single agent executing Algorithm 1. However, we note that, if we set δ = δ0/n, then, by a union bound, all n agents will have ˜d ∈ [(1 − )d , (1 + )d ]with probability δ0. The required running time t will depend just logarithmically on δ0and n.
Correctness of Encounter Rate in Expectation. The first step in proving Theorem 1 is to show that the encounter rate ˜d is an unbiased estimator of d . This result, in fact, holds for any ants randomly walking on any regular graph.
Lemma 2 (Unbiased Estimator). E ˜d = d.
Proof. We can decompose the collision bound c maintained
by each agent in Algorithm 1 as the sum of collisions with dif-ferent agents over difdif-ferent rounds. Specifically, give the n other agents arbitrary labels 1, 2, ..., n and let cj(r )equal 1 if the agent collides with agent j in round r , and 0 otherwise. By linearity of expectation, Ec =Pn
j =1 Pt
r =1Ecj(r ).
Since each agent is initially at a uniform random location and, after any number of steps, is still at a uniform random
loca-tion, for all j , r , Ecj(r ) = 1/A. Thus, Ec = nt/A = dt and E ˜d = Ec/t = d .
We note that the torus is bipartite, and hence two agents ini-tially located an odd number of steps away from each other will never meet via random walking. However, this fact does not change the expectation of ˜d computed above and, in fact, does not affect any of our following proofs.
With Lemma 2 in place, it remains to show that the encounter rate is close to its expectation with high probability and so pro-vides a good estimate of density. To do this, we must bound the strength of correlations between collisions of nearby agents in successive rounds, which can decrease the accuracy of the encounter rate-based estimate.
A Recollision Probability Bound.The key to bounding collision correlations is bounding the probability of a recollision between two agents in round r +m, assuming a collision in round r , which we do in this section.
Let cj=Ptr =1cj(r )be the total number of collisions with agent j . Due to the initial uniform distribution of the agents, the cj are all independent and identically distributed.
Each cj is the sum of highly correlated random variables; due to the slow mixing of the grid, if two agents collide at round r , they are much more likely to collide in successive rounds. However, by bounding this recollision probability, we are able to give strong moment bounds for the distribution of each cj. We bound not only its variance but all higher moments. This allows us to show that the average ˜d = 1/tPn
j =1cj
falls close to its expectation d with high probability, giving Theorem 1.
Lemma 3 (Recollision Probability Bound). Consider two agents a1 and a2randomly walking on a 2D torus of dimensions√A ×√A. If a1and a2collide in round r , for any m ≥ 0, the probability that a1and a2collide again in round r + m is Θ (1/m + 1) + O (1/A).
Fig. 2. A schematic of the proof of Lemma 3. We argue that the recollision probability of two agents after m steps (shown in red and blue) is equivalent to the probability that a length 2m random walk (shown in gray) returns to its origin. We then argue that the random walk is likely to take roughly m steps in both the x and y directions and hence has zero displacement in each direction with probability Θ(1/√m).
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Proof sketch. The full proof of Lemma 3 is given in SI
Appendix, section S2. We sketch the main ideas here and illus-trate them in Fig. 2.
We first show that the probability that a1 and a2 recollide in round r + m is identical to the probability that a single 2m-step random walk ends at its starting position.
The recollision probability is the probability that a1 and a2 have identical displacements after taking m steps each. By sym-metry of the random walk steps, this is equal to the probability that a1’s displacement vector is equal to the negative of a2’s. Fur-thermore, this is just the probability that their 2m total random walk steps have 0 overall displacement, which is the probability that a 2m-step random walk ends at its origin.
One idea might be to bound this “equalization probability” using the global mixing time of the torus (7). After Θ(A log A) steps, a random walk is nearly as likely to be at any node in the graph, including its origin. Thus, the equalization probability is bounded by O(1/A) for 2m = Ω(A log A). Unfortunately, such a bound says nothing about this probability for small m.
Thus, we must take a different approach. We first assume, for simplicity, that the walk is on an infinite grid, and so there is no possibility of returning to its origin by “wrapping around” the torus. We later show that this only affects the equalization prob-ability by an O (1/A) factor.
Considering a walk on the infinite grid, we condition on the walk taking roughly m steps in both the x and y directions, which occurs with high probability. We separately bound the probability of zero displacement in each direction.
It is well known that an m-step random walk on the line has roughly equal probability of ending at any point within radius Θ(√m)of its origin. It thus has probability Θ (1/√m)of end-ing at its origin. Fixend-ing the number of steps in each direction, the walk’s x and y displacements are independent. So, we can multi-ply the probabilities for each direction, giving the final bound of Θ (1/m + 1)(we write m + 1 in the denominator instead of m so that the formula holds for m = 0.)
Since it may be of independent interest, in Corollary 15 inSI Appendix, section S3, we restate the result of Lemma 3 explicitly in terms of a bound on the probability that a single random walk returns to its origin (equalizes) after m steps.
Collision Moment Bound. With Lemma 3 in hand, we can prove our collision moment bound, which we use to show that the num-ber of collisions an agent sees concentrates strongly around its expectation. We first show that any agent is likely to collide with many other agents during the execution of Algorithm 1, rather than repeatedly colliding with just a few other agents. That is, the probability that an agent collides at least once with any given other agent is not too low.
Lemma 4 (First Collision Probability). Assuming t ≤ A, for all j ∈ [1, ..., n], P [cj ≥ 1] = Θ (t /A log 2t ).
Proof sketch. By Lemma 3 and the assumption that t ≤ A, in t
rounds, an agent expects to recollide with any agent it encounters Pt −1
m=0Θ (1/m + 1) = Θ(log 2t )times. By Lemma 2, an agent expects to be involved in dt = nt/A total collisions. So, account-ing for recollisions, it expects to collide with Θ (nt/A log 2t) unique individuals. By symmetry, its collision probability with any single individual is thus Θ (t/A log 2t). A formal proof is given in SI Appendix, section S2.
Lemma 4 used that, by Lemma 3, an agent expects to collide O (log 2t )times with any other agent it encounters. We can, in fact, show that this bound is not just in expectation but extends to the higher moments of the collision distribution.
Lemma 5 (Collision Moment Bound).For j ∈ [1, ..., n], let ¯
cj def
= cj− Ecj and assume t ≤ A. There is some fixed constant wsuch that, for any integer k ≥ 2,
E h ¯ cjk i ≤tw k A · k ! log k −1 (2t ).
When k = 2, Lemma 5 gives a bound on the variance of cj, which can be used to show that cjfalls close to its mean with good prob-ability. By bounding the k th moment E[¯ck
j]for all k , we are able to show even stronger concentration results.
Proof sketch. Very roughly, we separately consider the simple
case when cj= 0and the case when cj≥ 1, whose probability is bounded in Lemma 4. In the latter case, we split cj over rounds as cj=Ptr =1cj(r )and expand out,
E[cjk] = t X r1=1 t X r2=1 ... t X rk=1 E [cj(r1)cj(r2)...cj(rk)]. [1]
E [cj(r1)cj(r2)...cj(rk)] is just the probability that two agents collide in each of rounds r1, r2, ..., rk. Assuming that r1≤ r2≤ ... ≤ rk and that there is a collision in round r1, we can apply Lemma 3 to bound this probability as ≤wk
/ (r2− r1+ 1)...(rk− rk −1+ 1)for some constant w .
Obtaining the theorem requires combining this bound with Eq. 1 and applying a number of careful rearrangements. How-ever, the bound on E [cj(r1)cj(r2)...cj(rk)] is the crux of the analysis. A full proof is inSI Appendix, section S2.
As with Lemma 3, the techniques used in Lemma 5 can be applied to bounding the moments of the number of equalizations of a single random walk. See Corollaries 16 and 17 inSI Appendix,
section S3.
Correctness of Encounter Rate with High Probability. Armed with Lemma 5, we can finally show thatPn
j =1cjconcentrates strongly about its expectation. Since ˜d = 1/tPn
j =1cj, this is enough to prove the accuracy of encounter rate-based density estima-tion (Algorithm 1). We first restate Lemma 5 using a standard “Bernstein condition” on the sumPn
j =1cj.
Corollary 6 (Bernstein Condition). Assuming t ≤ A,
E n X j =1 cj− E " n X j =1 cj #!k ≤ 1 2k !σ 2 bk −2
for all k ≥ 2 and some b = Θ(log 2t) and σ2
= Θ(td log 2t ).
Proof. By Lemma 5, there exists some constant w such that,
for σ2= wt log 2t /Aand b = w log 2t, ¯cj def = cj− Ecj satisfies E h ¯ cjk i ≤1 2k !σ 2 bk −2. Since each cj is independent,
E n X j =1 cj − E " n X j =1 cj #!k = E n X j =1 ¯ cj !k = n X j =1 E[¯ckj] ≤ n · k !σ2bk −2 2 .
The lemma follows after replacing σ2with nσ2= Θ(td log 2t ). We use the following concentration bound for random vari-ables satisfying such a Bernstein condition.
Lemma 7. Suppose that X satisfies E[(X − EX )k] ≤ 1/2k !σ2bk −2 for all k ≥ 3. Then, for any ∆ ≥ 0, P[|X − EX | ≥ ∆] ≤ 2e−∆2/2(σ2+b∆).
We conclude this section by proving our main theorem on the accuracy of random walk-based density estimation.
Proof of Theorem 1. In Algorithm 1, ˜d is set to 1/tPn
j =1cj.
So the probability that ˜dfalls within an multiplicative factor of its mean is the same as the probability thatPn
j =1cj falls within an multiplicative factor of its mean, which is equal to tE ˜d = td by Lemma 2. By Corollary 6 and Lemma 7,
δdef= P " n X j =1 cj− E " n X j =1 cj # ≥ E " n X j =1 cj ## = P " n X j =1 cj− td ≥ td # ≤ 2eΘ − 2 t 2 d 2 2(td log 2t +td log 2t ) . Restricting ≤ 1 and rearranging gives 2td /log 2t = Θ (log(1/δ))and so = Θplog(1/δ) log 2t/td, yielding the theorem.
Extensions to Other Topologies
We now discuss extensions of our results to a broader set of graph topologies, demonstrating the generality of our local mixing anal-ysis. We illustrate divergence between local and global mixing properties, which can have significant effects on random walk-based algorithms. Full proofs for all results in this section are deferred toSI Appendix, section S4.
From Recollision Bounds to Accurate Density Estimation. Our proofs for the 2D torus are largely independent of graph struc-ture, using just a recollision probability bound (Lemma 3) and the regularity (uniform node degrees) of the grid, so agents remain uniformly distributed on the nodes in each round (see, for example, Lemma 2). Hence, extending our results to other regular graphs primarily involves obtaining recollision probabil-ity bounds for these graphs.
We consider agents on a graph with A nodes that execute analogously to Algorithm 1, stepping to a random neighbor in each round. Again, we focus on the multiagent case, but similar bounds (resembling Corollaries 16 and 17 inSI Appendix, section
S3) hold for a single random walk. We start with a lemma which gives density estimation accuracy in terms of recollision proba-bility. This is a direct generalization of our grid analysis. Lemma 8 (Recollision Probability to Density Estimation Accuracy). Consider a regular graph with A nodes such that, if two ran-domly walking agents a1 and a2 collide in round r , for any 0 ≤ m ≤ t, the probability that they collide again in round r + m is Θ (β(m)) for some nonincreasing function β(m). Let B (t )def=Pt
m=0β(m). After running for t ≤ A steps, Algorithm 1 returns ˜dsuch that, for any δ > 0, with a probability of ≥1 − δ,
˜
d ∈ [(1 − )d , (1 + )d ]for = Oplog(1/δ)B(t)/td. Note that, in the special case of the 2D torus, by Lemma 3, we can set β(m) = 1/(m + 1) and hence B (t) = Θ(log 2t), recover-ing Theorem 1.
Density Estimation on k-Dimensional Tori. We first consider k-dimensional tori for general k . As k increases, local mixing becomes stronger, fewer recollisions occur, and density estima-tion becomes easier. In fact, for constant k ≥ 3, although the torus still mixes slowly, density estimation is as accurate as on the complete graph! Throughout this section, we assume that k is a small constant and so hide multiplicative factors in f (k ) for any function f in our asymptotic notation. We subscript the notation with k to make this clear. We begin with the case of k = 1. Lemma 9 (Recollision Probability Bound: Ring). If two randomly walking agents a1and a2are located on a 1D torus (a ring) with A nodes, and collide in round r , for any m ≥ 0, the probability that a1
and a2collide again in round r + m for k ≥ 1 is Θ 1/√m + 1 + O (1/A).
Proof sketch. This bound can be shown similarly to Lemma 3
(and, in fact, its proof is fully contained in the proof of Lemma 3.) A 2m-step random walk on a line ends at its origin with proba-bility Θ(1/√m + 1). On a ring with A nodes, the slightly weaker bound of Θ 1/√m + 1 + O (1/A) holds.
For m ≤ A, the O (1/A) term is absorbed into the Θ 1/√m + 1, and one can show that Pt
m=01/
√
m + 1 = Θ(√t ). Plugging into Lemma 8, on a ring, random walk-based density estimation gives = O
q log(1/δ)√t /td = O q log(1/δ)/√t d . Rearranging, t = Θ log(1/δ)/2d2 rounds are necessary to obtain a 1 ± approximation with a probability of ≥1 − δ for any , δ ∈ (0, 1). Local mixing on the ring is much worse than on the torus. Hence, density estimation is much more difficult, requiring t to be quadratic rather than linear in 1/d and 1/2.
We now cover k ≥ 3. While global mixing time is on the order of A2/k, local mixing is so strong that our accuracy bounds nearly match those of independent sampling.
Lemma 10 (Recollision Probability Bound: High-Dimensional Torus). If two randomly walking agents a1 and a2 are located on a k -dimensional torus with A nodes, and collide in round r , for any constant k ≥ 3, m ≥ 0, the probability that a1 and a2 collide in round r + m is Θk1/(m + 1)k /2+ O (1/A).
Proof sketch. The proof is similar to that of Lemma 3. To
col-lide in round r +m, the agents must have identical displacements in each of the k dimensions after m steps. Since k is a small con-stant, with high probability, the agents take Θ(m/k ) steps in each dimension. After conditioning on the step counts, the k collisions are independent, each occurring with probability Θ1/pm/k via the argument of Lemma 3. The result follows by multiplying these k probabilities together, noting that k dependence is hid-den in the asymptotic notation.
To convert the above bound to a density estimation accu-racy, we can use a slightly modified version of Lemma 8, which applies to the case when our collision probability is O (β(m)) but not neccesarily Θ(β(m)). For t ≤ A and k ≥ 3, Pt m=0(1/(m + 1) k /2 + 1/A) < 1+P∞ m=01/(m + 1) k /2 = O (1). So we can set B (t) = Ok(1)and have = Ok
plog(1/δ)/td. Rearranging, we require t = Θk log(1/δ)/2d. This matches independent sampling up to constants and multiplicative factors in k .
Density Estimation on Regular Expanders. When a graph does mix well globally, it mixes well locally. An obvious example is the complete graph, on which random walk-based and inde-pendent sampling-based density estimations are equivalent. We extend this intuition to any regular expander. An expander is a graph whose random walk matrix has its second eigenvalue bounded away from 1, and so on which random walks mix quickly. Expanders are “well-connected” graphs with many appli-cations, including in the design of robust communication net-works (16) and efficient sampling schemes (10).
Lemma 11 (Recollision Probability Bound: Regular Expander). Let G be a k -regular expander with A nodes and adjacency matrix M. Let W = 1/k · M be its random walk matrix, with eigenvalues λ1≥ λ2≥ ... ≥ λA. Let λ = max{|λ2|, |λA|} < 1. If two randomly walking agents a1and a2collide in round r , for any m ≥ 0, the prob-ability that they collide again in round r + m is, at most, λm+ 2/A.
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Proof sketch. The bound follows from noting that the stable
distribution on a regular expander is uniform, and the location distribution of any agent after m steps converges exponentially quickly to this distribution, with rate λ.
Again, we bound density estimation accuracy via a modifica-tion of Lemma 8, which applies when we have collision probabil-ity O(β(m)) but not necessarily Θ(β(m)). This modified lemma gives a B (t)2 dependence. B (t) = Pt
m=0β(m) ≤ 1/1 − λ + 2t /A. Assuming t = O(A), = O
q log(1/δ)/td (1 − λ)2 . Rearranging, t = Θ log(1/δ)/2 d (1 − λ)2, matching indepen-dent sampling up to a factor of O(1/(1 − λ)2).
Density Estimation k-Dimensional Hypercubes. Finally, we give bounds for a k -dimensional hypercube. Such a graph has A = 2k vertices mapped to the elements of {±1}k, with an edge between any two vertices that differ by Hamming distance 1. The hyper-cube is relatively fast mixing. Its adjacency matrix eigenvalues are [−k , −k + 2, ..., k − 2, k ]. Since it is bipartite, we can ignore the negative eigenvalues: To return to its origin, a random walk must take an even number of steps, so we need only need to con-sider the squared walk matrix W2, which has all positive eigenval-ues. Applying Lemma 11 with λ = Θ(1 − 2/k ) = Θ(1 − 1/ log A) gives t = Θ log (1/δ) log2(A)/2d. However, it is possible to remove the dependence on A via a more refined analysis: While the global mixing time of the graph increases as A grows, local mixing becomes stronger!
Lemma 12 (Recollision Probability Bound: k-Dimensional Hyper-cube). If two randomly walking agents a1and a2are located on a k-dimensional hypercube with A = 2kvertices and collide in round r, for any m ≥ 0, the probability that a1 and a2 collide in round r + mis O(7/10)m+ 1/√A.
Converting to a density estimation bound, we have B (t) = Pt
m=0β(m) ≤ 10/3 + t / √
A. If we assume t = O(√A), this gives = Oplog(1/δ)/td and so t = Θ log(1/δ)/2
d, matching independent sampling.
Applications
We conclude by discussing algorithmic applications of our ant-inspired density estimation algorithm (Algorithm 1), varia-tions on this algorithm, and the analysis techniques we have developed.
Social Network Size Estimation. Random walk-based density esti-mation is closely related to work on estimating the size of social networks and other massive graphs using random walks (12, 17–19). In these applications, one does not have access to the full graph (so cannot exactly count the nodes) but can simulate random walks by following links between nodes (20, 21). One approach is to run a single random walk and count repeat node visits (17, 18). Alternatively, ref. 12 proposes running multiple random walks and counting their collisions, which gives an esti-mate of the walk’s density. Since the number of walks is known, this yields an estimate for network size.
This approach can be significantly more efficient, since the dominant cost is typically in link queries to the network. With multiple, shorter random walks, this cost can be trivially dis-tributed to multiple servers simulating walks independently. Visit information can then be aggregated, and the collision count can be computed in a centralized manner.
Random walk-based algorithm for network size estimation.
Con-sider an undirected, connected, nonbipartite graph G = (V , E ). Let S be the set of vertices of G that are “known.” Initially, S = {v }, where v is a seed vertex. We can access G by
look-ing up the neighborhood Γ(vi)of any vertex vi∈ S and adding Γ(vi)to S .
To compute the network size |V |, we could scan S , looking up the neighbors of each vertex and adding them to the set. Repeat-ing this process until no new nodes are added ensures that S = V and we know the network size. However, this method requires |V | neighborhood queries. The goal is to use significantly fewer queries using random walk-based sampling.
A number of challenges are introduced by this application. While we can simulate many random walks on G, we can no longer assume these random walks start at randomly chosen nodes, as we do not have the ability to uniformly sample nodes from the network. Instead, we must allow the random walks to run for a burn-in phase of length proportional to the mixing time of G. After this phase, the walks are distributed approximately according to the stable distribution of G.
Further, in general, G is not regular. In the stable distribu-tion, a random walk is located at a vertex with probability pro-portional to its degree. Hence, collisions tend to occur more at higher-degree vertices. To correct for this bias, we count a colli-sion at vertex viwith weight 1/deg(vi).
Our results depend on a natural generalization of recollision probability. For any i , j , let p(vi, vj, m)be the probability that an m-step random walk starting at viends at vj. Define
β(m)def= maxi,jp(vi, vj, m) deg(vj)
.
Intuitively, β(m) is the maximum m-step collision probability, weighted by degree since higher-degree vertices are visited more in the stable distribution. Let B (t) =Pt
m=1β(m). Note that this weighted B (t) is trivially upper bounded by the unweighted mea-sure used in Lemma 8.
For simplicity, we initially ignore burn-in and assume that our walks start distributed exactly by the stable dis-tribution of G. A walk starts at vertex vi with probability pi
def
=deg(vi)/Pi deg(vi) =deg(vi)/2|E |, and initial locations are independent. We also assume knowledge of the average degree deg = 2|E |/|V |. SeeSI Appendix, section S5 for a rig-orous analysis of burn-in and average degree estimation.
Note that there are many ways to implement the count(·) func-tion used in Algorithm 2. One possibility is to simulate the ran-dom walks in parallel, recording their paths, and then to per-form centralized postprocessing to count collisions. As queries to the network are considered to dominate time cost, this colli-sion counting step is relatively inexpensive.
We prove the following theorem inSI Appendix, section S5. Theorem 13. If Algorithm 2 is run using n random walks for tsteps, as long as n2
t = ΘB (t )deg + 1/2
δ · |V |, then, with probability at least 1 − δ, it returns ˜A ∈ [(1 − )|V |, (1 + )|V |].
Proof sketch. The proof is similar to that of Theorem 1. It is not hard to see that, due to our reweighting of each collision by 1/deg(wj), EC = 1/|V |. The challenge is showing that C con-centrates around its expectation and hence ˜A = 1/C is close to |V |. Due to the complicating factors of nonuniform degree, we are unable to compute a general moment bound for each cj as done in Lemma 5. However, we can give a variance bound on C, and bound its deviation via Chebyshev’s inequality. This gives a worse dependence on the failure probability: 1/δ instead of log(1/δ). We note that this can be improved by running the algo-rithm log(1/δ) times, each with success probability 1/3, and tak-ing the median of the results.
Overall runtime and comparison to previous work. Let M denote
the burn-in time required before running Algorithm 2 (seeSI Appendix, section S5for details). To obtain a (1 ± ) estimate of network size with probability 1 − δ, we must run n random walks for M + t steps, making n(M + t) link queries, where, by Theorem 13 and our analysis of average degree estimation inSI Appendix, section S5, we have
n = Θ max deg degmin2δ , s |V | · (B (t ) deg + 1) t · 2δ . [2]
Typically, the second term dominates, since deg << |V |. Hence, by increasing t, we are able to use fewer random walks, signifi-cantly decreasing the number of link queries if M is large.
Ref. 12 uses a different approach, halting random walks and counting collisions immediately after burn-in. For reasonable node degrees, they require n = Θ|V | · deg/2
δ · q P deg(vi)2 . Assuming that q
P deg(vi)2< n, and setting t = 1, this is some-what smaller than our bound, asP deg(vi)2≥ |V | · deg. How-ever, Eq. 2 gives an important tradeoff: By increasing t, we can increase the number of steps in our random walks, decreasing the total number of walks.
As an illustrative example, consider a k -dimensional torus graph for k ≥ 3 [for k = 2, mixing time is Θ(|V |), so we might as well census the full graph]. The mixing time required for Algorithm 2 (see SI Appendix, section S5 for details) is M = Θ(log(|V |/δ)|V |2/k). All nodes have degree 2k , and using the bounds above, to obtain a (1 ± ) estimate of |V |, the algorithm of ref. 12 requires M · n = Θlog(|V |/δ)/√d · |V |2/k +1/2 link queries to obtain a size estimate. In contrast, assuming |V | is large, we require n = Θp|V |/t · 2δ, since, by Lemma 10, B (t) = O(1/k ) and deg = degmin= k. If we set t = Θ(M ), the total number of link queries needed is n(M + t) = Oplog(|V |/δ)/√d · |V |(k +1)/2k. This beats ref. 12 by improving dependence on |V | and the logarithmic burn-in term. Ignoring error dependences, if k = 3, ref. 12 requires Θ(n7/6
) queries, which is more expensive than fully censusing the graph. We require O(n2/3
)queries, which is sublinear in the graph size. We leave open comparing our bounds with those of ref. 12 on more natural classes of graphs. It would be interesting to deter-mine typical values of B (t) in real work networks or popular graph models, such as preferential attachment models and others with power-law degree distributions.
Distributed Density Estimation by Robot Swarms. Algorithm 1 can be directly applied as a simple and robust density estimation algorithm for robot swarms moving on a 2D plane modeled as a grid. Additionally, the algorithm can be used to estimate the frequency of certain properties within the swarm. Let d be the
overall population density and dP be the density of agents with some property P . Let fP= dP/dbe the relative frequency of P .
Assuming that agents with property P are distributed uni-formly in population and that agents can detect this property (through direct communication or some other signal), then they can separately track encounters with these agents. They can compute an estimate ˜d of d and ˜dP of dP. By Theorem 1, after running for t = Θ log(1/δ) log log(1/δ) log(1/d )/dP2
steps, with probability 1 − 2δ, d˜P/ ˜d ∈ [(1 − /1 + ) fP, (1 + /1 − ) fP] = [(1 − O ())fP, (1 + O ())fP]for small .
In an ant colony, properties may include whether an ant has recently completed a successful foraging trip (4), or if an ant is a nestmate or enemy (3). In a robotics setting, properties may include whether a robot is part of a certain task group, whether it has completed a certain task, or whether it has detected a certain event or environmental property.
Random Walk-Based Sensor Network Sampling. Finally, we believe our moment bounds for a single random walk (Corollaries 16 and 17 inSI Appendix, section S3) can be applied to random walk-based distributed algorithms for sensor network sampling. We leave obtaining rigorous bounds in this domain to future work. Random walk-based sensor network sampling (13, 14) is a technique in which a query message (a “token”) is initially sent by a base station to some sensor. The token is relayed randomly between sensors, which are connected via a grid network, and its value is updated appropriately at each step to give an answer to the query. This scheme is robust and efficient; it easily adapts to node failures and does not require setting up or storing spanning tree communication structures.
Random walk-based sampling could be used, for example, to estimate the percentage of sensors that have recorded a specific condition, or the average value of some measurement at each sensor. However, as in density estimation, unless an effort is made to record which sensors have been previously visited, addi-tional error is added due to repeat visits. Recording previous vis-its introduces computational burden: Either the token message size must increase or nodes themselves must remember which tokens they have seen. We are hopeful that our moment bounds can be used to show that this is unnecessary: Due to strong local mixing, the number of repeat sensor visits will be low, and the performance reduction limited.
We remark that estimating the percentage of sensors in a network or the density of robots in a swarm with a property that is uniformly distributed is a special case of a more gen-eral data aggregation problem: Each agent or sensor holds a value vi drawn independently from some distribution D. The goal is to estimate some statistic of D, such as its expectation. In the case of density estimation, vi is simply an indicator ran-dom variable which is 1 with probability d and 0 otherwise. Extending our results to more general data aggregation problems and showing that random walk sampling matches independent sampling in some cases is an interesting future direction. Discussion and Future Work
We have presented a theoretical analysis of random walk-based density estimation by agents moving synchronously on a 2D torus graph. We have also presented applications of our techniques to density estimation on other simple graph topologies and to the problems of social network size estimation and density estima-tion on robot swarms.
Aside from using our bounds to study sensor network sampling and giving improved theoretical and empirical understanding of our social network size estimation algorithm, our work leaves open a number of questions related to modeling random walk-based density estimation in ant colonies.
We feel that our simple computational model well reflects the behavior of ants estimating density via collision rates while
INAUGURAL ARTICLE COMPUTER SCIENCES SYSTEMS BIOLOGY
moving around a 2D surface. However, extending our results to more realistic models, e.g., with continuous movement along a surface which is either bounded or extends out indefinitely, is an interesting future direction.
As discussed, understanding how close actual ant movements are to random walks, and how nonrandom behavior influences density estimation via collision detection, is also important. In conjunction with this issue, removing our uniform density assumption and understanding how ants may estimate local pop-ulation densities which may vary throughout the nest or sur-rounding area is an important direction.
Finally, we note that the accuracy bound of Theorem 1 depends on the density d . In many applications, such as in quorum sens-ing, ants only need to detect when d is above some fixed thresh-old. In this case, better bounds, where t can be determined inde-pendently of the density, may be possible.
ACKNOWLEDGMENTS. We thank Amartya Shankha Biswas, Christopher
Musco, and Mira Radeva for useful discussions. Research was supported by National Science Foundation (NSF) Grants BIO-1455983 and CCF-1461559, NSF Center for Science of Information Grant CCF-0939370, and Air Force Office of Scientific Research Grant FA9550-13-1-0042. C.M. is partially sup-ported by an NSF graduate student fellowship.
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